A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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1answer
23 views

Calculate a differenciation

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
4
votes
1answer
34 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all k. Show that $S_{n}$ obeys the weak law of large numbers: ...
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0answers
16 views

Understanding a proof of the strong Markov property of Lévy processes

I don't understand the the last sentence of a proof of the Markov property for Lévy processes given in Jochen Wengenroth's textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008). I will appreciate ...
0
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0answers
11 views

zero drift brownian motions and barriers problem [duplicate]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > ...
2
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0answers
6 views

Techniques to prove FDD convergence

When examining a sequence of stochastic processes $(\textbf{X}_n)$, $n\geq1$ convergence of marginals, i.e. $\mathbf{X}_n(t)\to\mathbf{X}(t)$ (in distribution) is often not too hard to establish for ...
2
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1answer
26 views

Does Brownian Motion return to the origin infinitely soon?

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
2
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2answers
47 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
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0answers
16 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
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0answers
18 views

Arrival times of doubly stochastic process - Cox process

Let N(t) be a doubly stochastic process modeled with two independent processes $\tilde N(t)$, a Poisson process with rate 1 and an almost positive process $\lambda(t)$. We define $N(t)$ by $\tilde ...
0
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1answer
13 views

Probabilities in markov chain

I have problem with calculating the probability of Markov Chain with 3 states S = {0,1,2}. I need to calculate $P(X_1=1,X_2=1|X_0=2)$. In the answers to my workbook I am given solution: ...
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0answers
14 views

Ito Formula for Poisson Process: $d{X_t}=a_t dt +b_t dN_t$

Let $X_t$ solve the SDE $d{X_t}=a_t dt +b_t dN_t$, where $N_t$ is a Poisson Process. I want to demosntrate that in this case the Ito formula is the next one, but I dont know how to achieve it. ...
2
votes
2answers
43 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
0
votes
1answer
35 views

Ito Formula (Poisson basic process)

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, Ito´s fórmula is: $$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}a_t)dt + ...
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0answers
20 views

Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
1
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0answers
18 views

Limit of correlation function using transfer-matrix method

This question is about a stochastic process theory. I really very bad in this topic. That's why I have to ask for help. I may mistranslate some terms but I'll do my best to give you right information. ...
1
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1answer
25 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, what is the correct Ito´s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
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1answer
25 views

Notation: the $\sigma$-algebra $\mathcal{F}_\tau^+$

I'm reading a probability textbook on stochastic processes (Jochen Wengenroth's "Wahrscheinlichkeitstheorie", de Gruyter 2008) and the following notation: "$\mathcal{F}_\tau^+$" came up in the ...
0
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1answer
17 views

Aperiodicity in irreducible markov chains

I am stuck at aperiodic property of irreducible markov chain. Let us consider an irreducible markov chain. It's stated herein that for an irreducible markov chain, a single aperiodic state implies ...
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0answers
28 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
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0answers
23 views

Stochastic Differential Equation for Time Integral of Stochastic Process

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ ...
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1answer
26 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
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1answer
34 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
1
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1answer
25 views

Escape time for a not absorbing state

Let $X$ be a right-continuous Feller Dynkin process. For $r>0$ we define the $\{\mathcal{F}_t\}_t$ stopping time (which is called escape time) $$\eta_r=\inf\{t\geq 0: \|X_t -X_0\|\geq r\}$$ We have ...
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0answers
23 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
1
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1answer
19 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
0
votes
2answers
31 views

Two-dimensional Brownian motion

Let $B_1$ and $B_2$ be two $\mathbb{R}$-valued Brownian motions with $$\langle B_1,B_2\rangle=\int_0^t\rho_s ds,$$ where $\rho$ is progressively measurable with values in $(-1,1)$. We define ...
2
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1answer
36 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
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0answers
9 views

Stochastic process $X_t$, $X_t^2$, $X_{t^2}$ [on hold]

Can anyone explain me the difference between such stochastic processes:$X_t$, $X_t^2$, $X_{t^2}$ $X_t$ is let's say normal. How about two others? It's something to do with time, yes?
1
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1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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1answer
23 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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0answers
29 views

Square of a weakly stationary process

I have to prove that if $X_t$ is a weakly stationary process, $X_t^2$ is also. It is easy to prove the part referred to the means but I do not know how to work with covariances. Thanks!
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1answer
33 views

Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

I'm studying for exam and found this exercise which I don't really understand: Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process? So I need to show that ...
3
votes
1answer
35 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
votes
1answer
40 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
2
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0answers
23 views
+50

A non-trivial 2D SDE cannot have the same Joint Density as a 1D SDE

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
1
vote
1answer
36 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
2
votes
2answers
26 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
2
votes
1answer
35 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
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0answers
27 views

Independence of Poisson processes watched only some of the time

Let $(X_t)$ and $(Y_t)$ be independent homogeneous Poisson processes with rates $\lambda,\mu > 0$, and let $t_1, t_2, \dots$ and $t_1', t_2', \dots$ be two increasing sequences of possibly infinite ...
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votes
1answer
15 views

how to solve for Ut stochastic question [closed]

The process given by dUt = 􀀀-rUtdt + sigmadXt; U0 = u; where r,sigma are constants how to solve this equation for Ut? Thank you
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votes
1answer
15 views

What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero: ...
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0answers
10 views

What is “starting” static distribution?

I'm not sure if I call everything correctly in English in here, but i have a problem with stochastic processes - Markov chains to be more specific. I'm calculating the "starting" stationary ...
2
votes
1answer
22 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
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1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
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0answers
18 views

Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
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0answers
8 views

Class properties Markov chain [closed]

How can we show that an open class in a Markov chain is transient (both for finite and infinite)?
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1answer
16 views

Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration.

Let $\left(B_{t}\right)_{t\geq0}$ a Brownian motion. Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration. Remark: I try the following: It suffices to show ...
3
votes
0answers
33 views

No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
4
votes
1answer
54 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
-1
votes
1answer
31 views

What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [closed]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?